\(\int \frac {\sqrt {c (a x^2+b x^4)}}{d+e x^2} \, dx\) [279]

Optimal result

Integrand size = 27, antiderivative size = 105 \[ \int \frac {\sqrt {c \left (a x^2+b x^4\right )}}{d+e x^2} \, dx=\frac {\sqrt {a c x^2+b c x^4}}{e x}-\frac {\sqrt {b d-a e} \sqrt {a c x^2+b c x^4} \arctan \left (\frac {\sqrt {e} \sqrt {a+b x^2}}{\sqrt {b d-a e}}\right )}{e^{3/2} x \sqrt {a+b x^2}} \] Output:

(b*c*x^4+a*c*x^2)^(1/2)/e/x-(-a*e+b*d)^(1/2)*(b*c*x^4+a*c*x^2)^(1/2)*arcta 
n(e^(1/2)*(b*x^2+a)^(1/2)/(-a*e+b*d)^(1/2))/e^(3/2)/x/(b*x^2+a)^(1/2)
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {c \left (a x^2+b x^4\right )}}{d+e x^2} \, dx=\frac {c x \sqrt {a+b x^2} \left (\sqrt {e} \sqrt {a+b x^2}-\sqrt {b d-a e} \arctan \left (\frac {\sqrt {e} \sqrt {a+b x^2}}{\sqrt {b d-a e}}\right )\right )}{e^{3/2} \sqrt {c x^2 \left (a+b x^2\right )}} \] Input:

Integrate[Sqrt[c*(a*x^2 + b*x^4)]/(d + e*x^2),x]
 

Output:

(c*x*Sqrt[a + b*x^2]*(Sqrt[e]*Sqrt[a + b*x^2] - Sqrt[b*d - a*e]*ArcTan[(Sq 
rt[e]*Sqrt[a + b*x^2])/Sqrt[b*d - a*e]]))/(e^(3/2)*Sqrt[c*x^2*(a + b*x^2)] 
)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2087, 1466, 353, 60, 73, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {c \left (a x^2+b x^4\right )}}{d+e x^2} \, dx\)

\(\Big \downarrow \) 2087

\(\displaystyle \int \frac {\sqrt {a c x^2+b c x^4}}{d+e x^2}dx\)

\(\Big \downarrow \) 1466

\(\displaystyle \frac {\sqrt {a c x^2+b c x^4} \int \frac {x \sqrt {b c x^2+a c}}{e x^2+d}dx}{x \sqrt {a c+b c x^2}}\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {\sqrt {a c x^2+b c x^4} \int \frac {\sqrt {b c x^2+a c}}{e x^2+d}dx^2}{2 x \sqrt {a c+b c x^2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\sqrt {a c x^2+b c x^4} \left (\frac {2 \sqrt {a c+b c x^2}}{e}-\frac {c (b d-a e) \int \frac {1}{\sqrt {b c x^2+a c} \left (e x^2+d\right )}dx^2}{e}\right )}{2 x \sqrt {a c+b c x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\sqrt {a c x^2+b c x^4} \left (\frac {2 \sqrt {a c+b c x^2}}{e}-\frac {2 (b d-a e) \int \frac {1}{\frac {e x^4}{b c}+d-\frac {a e}{b}}d\sqrt {b c x^2+a c}}{b e}\right )}{2 x \sqrt {a c+b c x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\sqrt {a c x^2+b c x^4} \left (\frac {2 \sqrt {a c+b c x^2}}{e}-\frac {2 \sqrt {c} \sqrt {b d-a e} \arctan \left (\frac {\sqrt {e} \sqrt {a c+b c x^2}}{\sqrt {c} \sqrt {b d-a e}}\right )}{e^{3/2}}\right )}{2 x \sqrt {a c+b c x^2}}\)

Input:

Int[Sqrt[c*(a*x^2 + b*x^4)]/(d + e*x^2),x]
 

Output:

(Sqrt[a*c*x^2 + b*c*x^4]*((2*Sqrt[a*c + b*c*x^2])/e - (2*Sqrt[c]*Sqrt[b*d 
- a*e]*ArcTan[(Sqrt[e]*Sqrt[a*c + b*c*x^2])/(Sqrt[c]*Sqrt[b*d - a*e])])/e^ 
(3/2)))/(2*x*Sqrt[a*c + b*c*x^2])
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 1466

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbo 
l] :> Simp[(b*x^2 + c*x^4)^FracPart[p]/(x^(2*FracPart[p])*(b + c*x^2)^FracP 
art[p])   Int[x^(2*p)*(d + e*x^2)^q*(b + c*x^2)^p, x], x] /; FreeQ[{b, c, d 
, e, p, q}, x] &&  !IntegerQ[p]
 

rule 2087

Int[(u_)^(q_.)*(v_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^q*ExpandToSum 
[v, x]^p, x] /; FreeQ[{p, q}, x] && BinomialQ[u, x] && TrinomialQ[v, x] && 
 !(BinomialMatchQ[u, x] && TrinomialMatchQ[v, x])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(194\) vs. \(2(89)=178\).

Time = 0.33 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.86

method	result	size
pseudoelliptic	\(-\frac {-2 \sqrt {-\frac {c \left (a e -b d \right )}{d}}\, \sqrt {-d e}\, \sqrt {\left (b \,x^{2}+a \right ) c}+c \left (a e -b d \right ) \left (\ln \left (\frac {b c d x +\sqrt {-\frac {c \left (a e -b d \right )}{d}}\, \sqrt {\left (b \,x^{2}+a \right ) c}\, d -a c \sqrt {-d e}}{\sqrt {-d e}\, x +d}\right )-\ln \left (\frac {b c d x +\sqrt {-\frac {c \left (a e -b d \right )}{d}}\, \sqrt {\left (b \,x^{2}+a \right ) c}\, d +a c \sqrt {-d e}}{-\sqrt {-d e}\, x +d}\right )\right )}{2 \sqrt {-\frac {c \left (a e -b d \right )}{d}}\, \sqrt {-d e}\, e}\)	\(195\)
default	\(\frac {\sqrt {c \,x^{2} \left (b \,x^{2}+a \right )}\, \left (-\ln \left (-\frac {2 \left (\sqrt {-d e}\, b c x +\sqrt {\left (b \,x^{2}+a \right ) c}\, \sqrt {\frac {c \left (a e -b d \right )}{e}}\, e +a c e \right )}{-e x +\sqrt {-d e}}\right ) a c e +\ln \left (-\frac {2 \left (\sqrt {-d e}\, b c x +\sqrt {\left (b \,x^{2}+a \right ) c}\, \sqrt {\frac {c \left (a e -b d \right )}{e}}\, e +a c e \right )}{-e x +\sqrt {-d e}}\right ) b c d -\ln \left (-\frac {2 \left (\sqrt {-d e}\, b c x -\sqrt {\left (b \,x^{2}+a \right ) c}\, \sqrt {\frac {c \left (a e -b d \right )}{e}}\, e -a c e \right )}{e x +\sqrt {-d e}}\right ) a c e +\ln \left (-\frac {2 \left (\sqrt {-d e}\, b c x -\sqrt {\left (b \,x^{2}+a \right ) c}\, \sqrt {\frac {c \left (a e -b d \right )}{e}}\, e -a c e \right )}{e x +\sqrt {-d e}}\right ) b c d +2 \sqrt {\left (b \,x^{2}+a \right ) c}\, \sqrt {\frac {c \left (a e -b d \right )}{e}}\, e \right )}{2 x \sqrt {\left (b \,x^{2}+a \right ) c}\, e^{2} \sqrt {\frac {c \left (a e -b d \right )}{e}}}\)	\(335\)
risch	\(\frac {\sqrt {c \,x^{2} \left (b \,x^{2}+a \right )}}{e x}+\frac {\left (a e -b d \right ) \left (-\frac {\ln \left (\frac {\frac {2 c \left (a e -b d \right )}{e}+\frac {2 b c \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}{e}+2 \sqrt {\frac {c \left (a e -b d \right )}{e}}\, \sqrt {b c \left (x -\frac {\sqrt {-d e}}{e}\right )^{2}+\frac {2 b c \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}{e}+\frac {c \left (a e -b d \right )}{e}}}{x -\frac {\sqrt {-d e}}{e}}\right )}{2 e \sqrt {\frac {c \left (a e -b d \right )}{e}}}-\frac {\ln \left (\frac {\frac {2 c \left (a e -b d \right )}{e}-\frac {2 b c \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}{e}+2 \sqrt {\frac {c \left (a e -b d \right )}{e}}\, \sqrt {b c \left (x +\frac {\sqrt {-d e}}{e}\right )^{2}-\frac {2 b c \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}{e}+\frac {c \left (a e -b d \right )}{e}}}{x +\frac {\sqrt {-d e}}{e}}\right )}{2 e \sqrt {\frac {c \left (a e -b d \right )}{e}}}\right ) \sqrt {c \,x^{2} \left (b \,x^{2}+a \right )}\, \sqrt {\left (b \,x^{2}+a \right ) c}}{e x \left (b \,x^{2}+a \right )}\)	\(379\)

Input:

int((c*(b*x^4+a*x^2))^(1/2)/(e*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(-2*(-c/d*(a*e-b*d))^(1/2)*(-d*e)^(1/2)*((b*x^2+a)*c)^(1/2)+c*(a*e-b* 
d)*(ln((b*c*d*x+(-c/d*(a*e-b*d))^(1/2)*((b*x^2+a)*c)^(1/2)*d-a*c*(-d*e)^(1 
/2))/((-d*e)^(1/2)*x+d))-ln((b*c*d*x+(-c/d*(a*e-b*d))^(1/2)*((b*x^2+a)*c)^ 
(1/2)*d+a*c*(-d*e)^(1/2))/(-(-d*e)^(1/2)*x+d))))/(-c/d*(a*e-b*d))^(1/2)/(- 
d*e)^(1/2)/e
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.98 \[ \int \frac {\sqrt {c \left (a x^2+b x^4\right )}}{d+e x^2} \, dx=\left [\frac {x \sqrt {-\frac {b c d - a c e}{e}} \log \left (\frac {b^{2} c e^{2} x^{5} - 2 \, {\left (3 \, b^{2} c d e - 4 \, a b c e^{2}\right )} x^{3} + {\left (b^{2} c d^{2} - 8 \, a b c d e + 8 \, a^{2} c e^{2}\right )} x - 4 \, \sqrt {b c x^{4} + a c x^{2}} {\left (b e^{2} x^{2} - b d e + 2 \, a e^{2}\right )} \sqrt {-\frac {b c d - a c e}{e}}}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x}\right ) + 4 \, \sqrt {b c x^{4} + a c x^{2}}}{4 \, e x}, \frac {x \sqrt {\frac {b c d - a c e}{e}} \arctan \left (-\frac {\sqrt {b c x^{4} + a c x^{2}} {\left (b e x^{2} - b d + 2 \, a e\right )} \sqrt {\frac {b c d - a c e}{e}}}{2 \, {\left ({\left (b^{2} c d - a b c e\right )} x^{3} + {\left (a b c d - a^{2} c e\right )} x\right )}}\right ) + 2 \, \sqrt {b c x^{4} + a c x^{2}}}{2 \, e x}\right ] \] Input:

integrate((c*(b*x^4+a*x^2))^(1/2)/(e*x^2+d),x, algorithm="fricas")
 

Output:

[1/4*(x*sqrt(-(b*c*d - a*c*e)/e)*log((b^2*c*e^2*x^5 - 2*(3*b^2*c*d*e - 4*a 
*b*c*e^2)*x^3 + (b^2*c*d^2 - 8*a*b*c*d*e + 8*a^2*c*e^2)*x - 4*sqrt(b*c*x^4 
 + a*c*x^2)*(b*e^2*x^2 - b*d*e + 2*a*e^2)*sqrt(-(b*c*d - a*c*e)/e))/(e^2*x 
^5 + 2*d*e*x^3 + d^2*x)) + 4*sqrt(b*c*x^4 + a*c*x^2))/(e*x), 1/2*(x*sqrt(( 
b*c*d - a*c*e)/e)*arctan(-1/2*sqrt(b*c*x^4 + a*c*x^2)*(b*e*x^2 - b*d + 2*a 
*e)*sqrt((b*c*d - a*c*e)/e)/((b^2*c*d - a*b*c*e)*x^3 + (a*b*c*d - a^2*c*e) 
*x)) + 2*sqrt(b*c*x^4 + a*c*x^2))/(e*x)]
 

Sympy [F]

\[ \int \frac {\sqrt {c \left (a x^2+b x^4\right )}}{d+e x^2} \, dx=\int \frac {\sqrt {c x^{2} \left (a + b x^{2}\right )}}{d + e x^{2}}\, dx \] Input:

integrate((c*(b*x**4+a*x**2))**(1/2)/(e*x**2+d),x)
 

Output:

Integral(sqrt(c*x**2*(a + b*x**2))/(d + e*x**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c \left (a x^2+b x^4\right )}}{d+e x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*(b*x^4+a*x^2))^(1/2)/(e*x^2+d),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (89) = 178\).

Time = 0.13 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.73 \[ \int \frac {\sqrt {c \left (a x^2+b x^4\right )}}{d+e x^2} \, dx=-\frac {{\left (b c d \mathrm {sgn}\left (x\right ) - a c e \mathrm {sgn}\left (x\right )\right )} \arctan \left (\frac {\sqrt {b c x^{2} + a c} e}{\sqrt {b c d e - a c e^{2}}}\right )}{\sqrt {b c d e - a c e^{2}} e} + \frac {\sqrt {b c x^{2} + a c} \mathrm {sgn}\left (x\right )}{e} + \frac {{\left (b c d \arctan \left (\frac {\sqrt {a c} e}{\sqrt {b c d e - a c e^{2}}}\right ) - a c e \arctan \left (\frac {\sqrt {a c} e}{\sqrt {b c d e - a c e^{2}}}\right ) - \sqrt {b c d e - a c e^{2}} \sqrt {a c}\right )} \mathrm {sgn}\left (x\right )}{\sqrt {b c d e - a c e^{2}} e} \] Input:

integrate((c*(b*x^4+a*x^2))^(1/2)/(e*x^2+d),x, algorithm="giac")
 

Output:

-(b*c*d*sgn(x) - a*c*e*sgn(x))*arctan(sqrt(b*c*x^2 + a*c)*e/sqrt(b*c*d*e - 
 a*c*e^2))/(sqrt(b*c*d*e - a*c*e^2)*e) + sqrt(b*c*x^2 + a*c)*sgn(x)/e + (b 
*c*d*arctan(sqrt(a*c)*e/sqrt(b*c*d*e - a*c*e^2)) - a*c*e*arctan(sqrt(a*c)* 
e/sqrt(b*c*d*e - a*c*e^2)) - sqrt(b*c*d*e - a*c*e^2)*sqrt(a*c))*sgn(x)/(sq 
rt(b*c*d*e - a*c*e^2)*e)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c \left (a x^2+b x^4\right )}}{d+e x^2} \, dx=\int \frac {\sqrt {c\,\left (b\,x^4+a\,x^2\right )}}{e\,x^2+d} \,d x \] Input:

int((c*(a*x^2 + b*x^4))^(1/2)/(d + e*x^2),x)
 

Output:

int((c*(a*x^2 + b*x^4))^(1/2)/(d + e*x^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {c \left (a x^2+b x^4\right )}}{d+e x^2} \, dx=\frac {\sqrt {c}\, \left (-\sqrt {e}\, \sqrt {-a e +b d}\, \mathit {atan} \left (\frac {\sqrt {b}\, \sqrt {b \,x^{2}+a}\, e x +a e +b e \,x^{2}}{\sqrt {e}\, \sqrt {b \,x^{2}+a}\, \sqrt {-a e +b d}+\sqrt {e}\, \sqrt {b}\, \sqrt {-a e +b d}\, x}\right )+\sqrt {b \,x^{2}+a}\, e \right )}{e^{2}} \] Input:

int((c*(b*x^4+a*x^2))^(1/2)/(e*x^2+d),x)
 

Output:

(sqrt(c)*( - sqrt(e)*sqrt( - a*e + b*d)*atan((sqrt(b)*sqrt(a + b*x**2)*e*x 
 + a*e + b*e*x**2)/(sqrt(e)*sqrt(a + b*x**2)*sqrt( - a*e + b*d) + sqrt(e)* 
sqrt(b)*sqrt( - a*e + b*d)*x)) + sqrt(a + b*x**2)*e))/e**2